{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 07 数据归一化处理"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 最值归一化 Normalization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = np.random.randint(0, 100, 100) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "array([34,  9, 26, 98, 82,  8,  5, 77, 27, 74,  1, 12, 66, 45, 84,  8, 74,\n       62,  0, 55, 73, 10, 21, 36,  3, 58, 44, 24, 43, 29, 79, 22,  5, 69,\n       43, 73, 71, 49, 95, 11, 79,  5,  1, 31, 30, 81, 68, 57, 14, 77, 98,\n       23, 23, 93, 17, 91,  1, 12, 61, 34, 99, 27, 15, 37, 66, 67, 21, 17,\n       18, 75, 91, 43, 63, 27, 16, 49, 98, 25, 22, 94, 44, 66, 78, 54, 42,\n       36, 19, 89, 49,  2, 66, 81, 18, 83, 54, 93,  3, 61, 12, 36])"
     },
     "metadata": {},
     "execution_count": 31
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "array([0.34343434, 0.09090909, 0.26262626, 0.98989899, 0.82828283,\n       0.08080808, 0.05050505, 0.77777778, 0.27272727, 0.74747475,\n       0.01010101, 0.12121212, 0.66666667, 0.45454545, 0.84848485,\n       0.08080808, 0.74747475, 0.62626263, 0.        , 0.55555556,\n       0.73737374, 0.1010101 , 0.21212121, 0.36363636, 0.03030303,\n       0.58585859, 0.44444444, 0.24242424, 0.43434343, 0.29292929,\n       0.7979798 , 0.22222222, 0.05050505, 0.6969697 , 0.43434343,\n       0.73737374, 0.71717172, 0.49494949, 0.95959596, 0.11111111,\n       0.7979798 , 0.05050505, 0.01010101, 0.31313131, 0.3030303 ,\n       0.81818182, 0.68686869, 0.57575758, 0.14141414, 0.77777778,\n       0.98989899, 0.23232323, 0.23232323, 0.93939394, 0.17171717,\n       0.91919192, 0.01010101, 0.12121212, 0.61616162, 0.34343434,\n       1.        , 0.27272727, 0.15151515, 0.37373737, 0.66666667,\n       0.67676768, 0.21212121, 0.17171717, 0.18181818, 0.75757576,\n       0.91919192, 0.43434343, 0.63636364, 0.27272727, 0.16161616,\n       0.49494949, 0.98989899, 0.25252525, 0.22222222, 0.94949495,\n       0.44444444, 0.66666667, 0.78787879, 0.54545455, 0.42424242,\n       0.36363636, 0.19191919, 0.8989899 , 0.49494949, 0.02020202,\n       0.66666667, 0.81818182, 0.18181818, 0.83838384, 0.54545455,\n       0.93939394, 0.03030303, 0.61616162, 0.12121212, 0.36363636])"
     },
     "metadata": {},
     "execution_count": 32
    }
   ],
   "source": [
    "# 最值归一化\n",
    "(x - np.min(x)) / (np.max(x) - np.min(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X = np.random.randint(0, 100, (50, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 因为np.array只能存储一种数据，所以要进行类型的转化\n",
    "X = np.array(X, dtype=float)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "array([[40.,  7.],\n       [48., 68.],\n       [41., 31.],\n       [90., 58.],\n       [78., 36.],\n       [59., 97.],\n       [ 0., 36.],\n       [54., 88.],\n       [83.,  4.],\n       [94., 63.]])"
     },
     "metadata": {},
     "execution_count": 35
    }
   ],
   "source": [
    "X[:10,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 每一个列进行转化\n",
    "X[:,0] = (X[:,0] - np.min(X[:,0])) / (np.max(X[:,0]) - np.min(X[:,0]))\n",
    "X[:,1] = (X[:,1] - np.min(X[:,1])) / (np.max(X[:,1]) - np.min(X[:,1]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "array([[0.40816327, 0.07216495],\n       [0.48979592, 0.70103093],\n       [0.41836735, 0.31958763],\n       [0.91836735, 0.59793814],\n       [0.79591837, 0.37113402],\n       [0.60204082, 1.        ],\n       [0.        , 0.37113402],\n       [0.55102041, 0.90721649],\n       [0.84693878, 0.04123711],\n       [0.95918367, 0.64948454]])"
     },
     "metadata": {},
     "execution_count": 37
    }
   ],
   "source": [
    "X[:10,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "plt.scatter(X[:,0], X[:,1])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "0.4616326530612245"
     },
     "metadata": {},
     "execution_count": 39
    }
   ],
   "source": [
    "np.mean(X[:,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "0.27299349393366956"
     },
     "metadata": {},
     "execution_count": 40
    }
   ],
   "source": [
    "np.std(X[:,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "0.4404123711340206"
     },
     "metadata": {},
     "execution_count": 41
    }
   ],
   "source": [
    "np.mean(X[:,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "0.2898212275283304"
     },
     "metadata": {},
     "execution_count": 42
    }
   ],
   "source": [
    "np.std(X[:,1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 均值方差归一化 Standardization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X2 = np.random.randint(0, 100, (50, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X2 = np.array(X2, dtype=float)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "array([[80.,  1.],\n       [ 0., 99.],\n       [63.,  1.],\n       [64., 25.],\n       [50., 72.],\n       [57., 90.],\n       [77., 37.],\n       [74., 15.],\n       [71., 74.],\n       [63., 36.]])"
     },
     "metadata": {},
     "execution_count": 45
    }
   ],
   "source": [
    "X2[:10,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# 均值方差归一化，使用numpy中的std函数\n",
    "X2[:,0] = (X2[:,0] - np.mean(X2[:,0])) / np.std(X2[:,0])\n",
    "X2[:,1] = (X2[:,1] - np.mean(X2[:,1])) / np.std(X2[:,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "plt.scatter(X2[:,0], X2[:,1])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "-1.1102230246251565e-16"
     },
     "metadata": {},
     "execution_count": 48
    }
   ],
   "source": [
    "np.mean(X2[:,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "0.9999999999999998"
     },
     "metadata": {},
     "execution_count": 49
    }
   ],
   "source": [
    "np.std(X2[:,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "-1.021405182655144e-16"
     },
     "metadata": {},
     "execution_count": 50
    }
   ],
   "source": [
    "np.mean(X2[:,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "1.0"
     },
     "metadata": {},
     "execution_count": 51
    }
   ],
   "source": [
    "np.std(X2[:,1])"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3.6.8 64-bit ('anaconda': conda)",
   "language": "python",
   "name": "python36864bitanacondaconda085db2e8b14649f4b32196068f7124e9"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.7-final"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}